January 22 2015

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Method

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Choose three squares A,B and X anywhere on the Sudoku grid.

Squares A,B and X can have any number of candidates.

Square A and square B must share at least one common candidate.

Choose a common candidate.

Perform a test with the chosen common candidate in square A set to 'off' and note the result obtained in square X.

Perform a test with the chosen common candidate in square B set to 'off' and note the result obtained in square X.

If the two results are not incompatible then the tests are inconclusive.

Choose a different candidate or different squares and try again.

If the two results are incompatible...

It's telling us that the puzzle won't complete with square A and/or square B set with their common factor in these states.

It's telling us that to complete the puzzle we would need to choose one of these three options.

i Change the state of the chosen common candidate in square A from 'off' to 'on'.

ii Change the state of the chosen common candidate in square B from 'off' to 'on'.

iii Change the state of the chosen common candidate in both square A and square B from 'off' to 'on'.

This third option is not applicable if square A and square B are in the same row or column or box.

What we don't know...

We don't know which option to choose.

What we do know...

We know that whichever option we chose...

At least one of the two squares A and B would then contain the chosen common candidate.

What's the point?

Any square on the grid that can see both square A and square B may have the chosen common candidate eliminated.

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Examples of incompatible results with a chosen common candidate 4

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X=12345678

"The value of square X is 7 when A is not 4. The value of square X is 8 when B is not 4"

X=12345678

"The value of square X is 7 when A is not 4. The value of square X is not 7 when B is not 4"

X=12345678

"The value of square X is 1 or 2 when A is not 4. The value of square X is 3 or 4 when B is not 4"

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Real world examples

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- Code: Select all
`*--------------------------------------------------------------------*`

|*18 7 6 | 2 13 9 | 4 5 *38 |

|-12458 3 14589 |X47 1457 A14 |*78 2789 6 |

| 245 259 459 | 467 34567 8 | 37 279 1 |

|----------------------+----------------------+----------------------|

| 1458 1568 1458 | 3 9 46 | 1578 4678 2 |

| 3 568 2 | 1 468 7 | 9 468 458 |

| 7 1689 1489 | 468 2 5 | 138 468 348 |

|----------------------+----------------------+----------------------|

| 6 1258 1358 | 9 13478 1234 | 578 478 4578 |

| 9 258 358 | 478 3478 234 | 6 1 4578 |

|B18 4 7 | 5 168 -16 | 2 3 9 |

*--------------------------------------------------------------------*

A=r2c6, B=r9c1, X=r2c4

A=4 => X=r2c4 = 7

B=8 => r1c9=8 -> r2c7=7 -> X=r2c4 = 4

=> A or B is 1, r2c1,r9c6<>1

- Code: Select all
`.------------------.------------------.-------------.`

| 2467 24 5 | 3 2467 8 | 679 1 679 |

| 2367 29 2369 | 127 1267 126 | 8 5 4 |

| 467 1 8 | 5 9 46 | 3 2 67 |

:------------------+------------------+-------------:

| 45 3 7 | 8 124 124 | 59 6 159 |

| 9 45 X16 | A17 13467 1346 | 5-7 8 2 |

| *126 8 *126 | 9 16-7 5 | 4 3 B17 |

:------------------+------------------+-------------:

| 13 6 13 | 4 5 9 | 2 7 8 |

| 8 7 4 | 12 123 123 | 56 9 56 |

| 25 259 29 | 6 8 7 | 1 4 3 |

.------------------.------------------.-------------.

A=r5c4, B=r6c9,X=r5c3

A=1->X=6

B=1->X=1

=> A or B is 7, r5c7,r6c5<>7

- Code: Select all
`.-----------------.------------------.------------------.`

| 9 13 2 | 5 6 38 | 48 7 1348 |

| 4 8 67 | 137 127 237 | 9 2356 35 |

| 167 137 5 | 9 4 2378 | 268 236 138 |

:-----------------+------------------+------------------:

| 5678 4 678 | 68 279 25679 | 1 235 358 |

| 2 9 178 | 148 3 B57 | 478 *45 6 |

| 15678 57 3 | 1468 127 2567 | 2478 9 458 |

:-----------------+------------------+------------------:

| 78 2 48 | A67 5 1 | 3 X46 9 |

| 157 157 147 | 367 79 369-7 | 46 8 2 |

| 3 6 9 | 2 8 4 | 5 1 7 |

'-----------------'------------------'------------------'

A=r7c4, B=r5c6, X=r7c8

A=6->X=4

B=5->X=6

=> A or B is 7, r8c6<>7

- Code: Select all
`.---------------.-------------.-------------.`

| 5 4 3 | 7 2 9 | 1 6 8 |

| 2 1 b69 | 4 8 a36 | 9-5 7 B35 |

| 68 89 7 | 1 5 36 | 49 2 34 |

:---------------+-------------+-------------:

| 7 59 c29 | 259 1 4 | 3 8 6 |

| 68 589 469 | 59 3 7 | 2 1 45 |

| 1 3 X24 | 25 6 8 | A45 9 7 |

:---------------+-------------+-------------:

| 9 7 8 | 3 4 2 | 6 5 1 |

| 4 2 5 | 6 7 1 | 8 3 9 |

| 3 6 1 | 8 9 5 | 7 4 2 |

'---------------'-------------'-------------'

A=r6c7, B=r2c9, X=r6c3

A=4->X=2

B=3->X=4

=> A or B is 5, r2c7<>5

- Code: Select all
`*--------------------------------------------------*`

| 5 4 3 | 7 2 9 | 1 6 8 |

| 2 1 *69 | 4 8 36 | 59 7 35 |

| 68 89 7 | 1 5 36 | B49 2 3-4 |

*----------------+----------------+----------------|

| 7 59 29 | 259 1 4 | 3 8 6 |

| 68 589 X469 | 59 3 7 | 2 1 A45 |

| 1 3 24 | 25 6 8 | 5-4 9 7 |

*----------------+----------------+----------------|

| 9 7 8 | 3 4 2 | 6 5 1 |

| 4 2 5 | 6 7 1 | 8 3 9 |

| 3 6 1 | 8 9 5 | 7 4 2 |

*--------------------------------------------------*

A=r5c9, B=r3c7, X=r5c3

A=5->X=4

B=9=>r2c3=9->X=6

=> A or B is 4, r3c9,r6c7<>4

- Code: Select all
`.------------------.---------------.------------------.`

| 8 3 9 | 45-6 2 X45 | 156 B16 7 |

| 1 6 2 | a358 7 b358 | 359 389 4 |

| 45 47 457 | A368 9 1 | 3-6 238-6 23 |

:------------------+---------------+------------------:

| 49 2 134 | 149 8 6 | 1349 7 5 |

| 7 1489 1348 | 149 5 249 | 1349 1239 6 |

| 469 5 146 | 7 3 e249 | 8 c129 d29 |

:------------------+---------------+-------- ---------:

| 569 1789 15678 | 2 4 35789 | 3679 369 39 |

| 3 4789 478 | 89 6 789 | 2 5 1 |

| 2 79 567 | 359 1 3579 | 3679 4 8 |

'------------------'---------------'------------------'

A=r3c4, B=r1c8, X=r1c6

A<>6=>r3c4,r2c46=358->X=4

B<>6=>r6c89=29=>r6c6=4->X=5

=> A or B is 6, r1c4,r3c7,r3c8<>6

- Code: Select all
`.---------------.----------.------------.`

| 3 5 2 | 4 7 1 | 6 9 8 |

| 7 89 89 | 6 3 5 | 14 14 2 |

| 6 4 1 | 2 9 8 | 5 3 7 |

:---------------+----------+------------:

| 5 1 3 | 7 4 2 | 9 8 6 |

| 8 X27 47 | 5 6 9 | 3 24 1 |

| 9 A26 46 | 1 8 3 | 24 7 5 |

:---------------+----------+------------:

| 2 d79-6 679 | 8 15 4 | 17 B56 3 |

| 4 c78 5 | 3 12 6 | 78 12 9 |

| 1 3 b68 | 9 25 7 | 28 a56 4 |

'---------------'----------'------------'

A=r6c2, B=r7c8, X=r5c2

A<>6->X=7

B<>6=>r9c8=6=>r9c3=8=>r8c2=7->X=2

=> A or B is 6, r7c2<>6